Journal of Number Theory 80, 260272 (2000)
Article ID jnth.1999.2451, available online at http:www.idealibrary.com on
A Lower Bound for the Height in Abelian Extensions
Francesco Amoroso
Dipartimento di Matematica, Universita di Torino,
Via Carlo Alberto 10, 10123 Torino, Italy
and
Roberto Dvornicich
Dipartimento di Matematica, Universita di Pisa,
Via F. Buonarroti 2, 56127 Pisa, Italy
Communicated by M. Waldschmidt
Received December 13, 1998
We produce an absolute lower bound for the height of the algebraic numbers
(different from zero and from the roots of unity) lying in an abelian extension of
the rationals. The proof rests on elementary congruences in cyclotomic fields and
on KroneckerWeber theorem. 2000 Academic Press
1. INTRODUCTION
Using Weil's height (the logarithmic and absolute height), the famous
Lehmer problem [Le 1933] reads as follows: does there exist a positive
constant c such that for any algebraic number :, :{0 and : not a root of
unity, we have
h(:)
c
?
[Q(:) : Q]
(1)
This problem is still open, the best unconditional lower bound in this direction being a theorem of Dobrowolski [Do 1979]. However, in some special
cases not only inequality (1) is true, but it can also be sharpened. Assume
for instance that L is a ``kroneckerian field'' (i.e. either a totally real number
field or a totally complex quadratic extension of such a field); then, as a
special case of a more general result, Schinzel proved that
1+- 5
1
r0.2406
h(:) log
2
2
260
0022-314X00 35.00
Copyright 2000 by Academic Press
All rights of reproduction in any form reserved.
(2)
LOWER BOUND FOR THE HEIGHT IN ABELIAN EXTENSIONS
261
if : # L* and |:| {1 (apply [Sc 1973], Corollary 1', p. 386, to the linear
polynomial P(z)=z&:). In particular, by Kronecker's theorem, this
inequality holds if : is an algebraic integer different from zero and from the
roots of unity. Although this additional assumption makes no harm if one
is interested in proving inequality (1) (since the Weil's height of a noninteger : is trivially (log 2)[(Q(:) : Q]), it may happen that non-integers
in kroneckerian fields have Weil's height smaller than 12 log(1+- 5)2. For
instance, the roots of the irreducible polynomial 2x 4 &3x 2 +2 # Z[x] belong
to an abelian extension and have absolute value 1, hence their height is
(log 2)4r0.1732.
When the extension LQ is Galois, L is kroneckerian if and only if the
complex conjugation lies in the center of the Galois group. In this situation
there are other proofs of (2) (see for instance [La 1980] and [Fl 1994]),
all of these relying on the following idea: ;=:: is totally positive and {1,
otherwise all of its conjugates would have absolute value 1 and then, since
: is an integer, it would be a root of unity. So the condition that : is an
integer is essential for this kind of argument.
Assume further LQ abelian. In the paper quoted above [La 1980],
Laurent uses the idea that the Frobenius automorphism relative to a prime
ideal depends only on the underlying rational prime to prove that
h(:)c(L), where c(L) depends on the smallest non-ramified rational
prime in L (although the proof is given explicitly only for integers, it can
be easily adapted to the general case). In this paper we introduce a new
idea which allows us to deal with ramified primes, thereby leading to an
absolute lower bound for the height in abelian extensions. Our main result
is the following:
Theorem. Let LQ be an abelian extension and let : # L*, : not a root
of unity. Then
h(:)
log 5
r0.1341.
12
We do not know that is the best possible lower bound, but certainly the
constant (log 5)12 cannot be replaced by any number >(log 7)12 (see
the example after the remarks following the proof of Corollary 2).
As a consequence of inequalities which hold also for non integers :, our
method enables us to prove some results on lower bounds for the norm
and the class group of abelian fields.
Notation. For a natural number m3 we denote by ` m a primitive
m-root of unity and we let K m =Q(` m ) be the m-th cyclotomic field of
262
AMOROSO AND DVORNICICH
degree ,(m) over Q. Since K m =K 2m for odd m, we shall assume that m2
(mod 4).
2. CONGRUENCES AND PROOF OF THE MAIN RESULT
Lemma 1. Let K be a number field and let & be a non-archimedean place
of K. Then, for any : # K*, there exists an algebraic integer ; such that ;:
is also integer and
| ;| & =max[1, |:| & ] &1.
Proof. Let 7 0 be the set of non-archimedean places w of K such that
max[1, |:| w ]>1. Let 7=7 0 _ [&] and choose an arbitrary non-archimedean
place w 0 . By the ``strong approximation theorem'' (see [CF], Chapter II,
Section 15, p. 67), there exists ; # K such that | ;&: &1 | w <max[1, |:| ] &1
for any w # 7 and |;| w 1 if w  7 _ [w 0 ]. Using the ultrametric inequality,
we deduce that
| ;| w =max[1, |:| w ] &1
for any w # 7 and | ;| w 1 if w  7 _ [w 0 ]. Therefore, | ;| w 1 and
|:;| w 1 for any finite place w (and so ; and :; are both integers) and
| ;| & =max[1, |:| & ] &1, since & # 7. K
Lemma 2. Let p be a rational prime. Then there exists _=_ p # Gal(K m Q)
with the following two properties.
(1)
If p |3 m, then
p | (# p &_#)
for any integer # # K m .
(2)
If p | m, then
p | (# p &_# p )
for any integer # # K m . Moreover, if _# p =# p for some # # K m , then there
exists a root of unity ` # K m such that `# is contained in a proper cyclotomic
subextension of K m .
Proof. Assume first that p |3 m. Let _ # Gal(K m Q) be defined by _` m =` mp .
For any integer # # K m we have #= f (` m ) for some f # Z[x]; hence
# p # f (` mp )# f (_` m )#_#
(mod p).
LOWER BOUND FOR THE HEIGHT IN ABELIAN EXTENSIONS
263
Assume now that p | m. The Galois group Gal(K m K m p ) is cyclic of
order k= p or k= p&1 depending on whether p 2 | m or not. Let _ be one
of its generators; hence _` m =` p ` m for some primitive p-root of unity ` p .
For any integer #= f (` m ) # Z[` m ], we have
# p # f (` mp )# f (_` mp )#_# p(mod p).
Suppose finally that _# p =# p: then _#=` up # for some integer u. It follows
that _(#` um )=#` um , hence #` um belongs to the fixed field K m p , as desired.
K
Proposition 1. Let p3 be a prime number and let : # K*
m , : not a root
of unity. Then
log( p2)
,
p+1
h(:)
log( p2)
,
2p
{
if p |3 m;
otherwise.
Proof. Let K=K m and let & be a place of K dividing p (with | p| & =1p).
By Lemma 1, there exists an integer ; # K such that :; is integer and
| ;| & =max[1, |:| & ] &1.
Let _=_ p the homomorphism given by Lemma 2. Assume first that p |3 m;
then
|(:;) p &_(:;)| & 1
p
and
1
| ; p &_;| & .
p
Using the ultrametric inequality, we deduce that
p
|: p &_:| & = | ;| &
|(:;) p &_(:;)+(_;&; p ) _:| &
&
p
| ;| &
max( |(:;) p &_(:;)| & , | ; p &_;| & |_:| & )
&
1
max(1, |:| & ) p max(1, |_:| & ).
p
Moreover : p {_:, since : is not a root of unity (see for instance [Do],
Lemma 2i)). We now apply the product formula to #=: p &_:.
264
AMOROSO AND DVORNICICH
[K & : Q & ]
[K & : Q & ]
[K & : Q & ]
log |#| & + :
log |#| & + :
log |#| &
[K
:
Q]
[K
:
Q
]
[K : Q]
&
&
& |3 &|p
&|
0= :
& |3 p
[K & : Q & ]
[K & : Q & ]
( p log + |:| & +log + |_:| & )& :
log p
[K : Q]
[K : Q]
& |3 & |p
:
[K & : Q & ]
( p log + |:| & +log + |_:| & +log 2)
[K : Q]
&|
+:
= ph(:)+h(_:)&log p+log 2
=( p+1) h(:)&log( p2).
Therefore,
h(:)
log( p2)
.
p+1
Assume now that p | m: again by Lemma 2, we have
|(:;) p &_(:;) p | & 1
p
and
1
| ; p &_; p | & .
p
Using the ultrametric inequality, we find
p
|: p &_: p | & = | ;| &
|(:;) p &_(:;) p +(_; p &; p ) _: p | &
&
1
max(1, |:| & ) p max(1, |_:| & ) p.
p
Moreover, we can assume : p {_: p. Otherwise, again by Lemma 2, there
exists a root of unity ` # K such that `: is contained in a proper cyclotomic
subextension of K; hence h(:)=h(`:) and, by induction, h(`:)log( p2)(2p).
Applying the product formula to #=: p &_: p as in the first part of the
proof, we get
0 ph(:)+ ph(_:)&log p+log 2=2ph(:)&log( p2).
Therefore, in this case,
h(:)
log( p2)
.
2p
K
LOWER BOUND FOR THE HEIGHT IN ABELIAN EXTENSIONS
265
The previous proposition gives, via KroneckerWeber's theorem, the
lower bound
h(:)
log(52)
r0.09163
10
for the height of a non-zero algebraic number : (: not a root of unity)
lying in an abelian extension. To reach the better lower bound announced
in the introduction, we study the special case p=2:
Lemma 3.
(1)
Let _=_ 2 be as before, and let # # K m be an integer.
If 4 |3 m, then
4 | (# 4 &_# 2 ).
(2)
If 4 | m, then
4 | (# 2 &_# 2 ).
Proof.
The first assertion immediately follows from Lemma 2, since
# 4 &_# 2 =(# 2 &_#) } ((&#) 2 &_(&#)).
For the second, let #= i a i ` im be an integer of K m . Then _#= i (&1) i a i ` im ,
hence _#&# and _#+# are both divisible by 2, so _# 2 &# 2 is divisible by 4. K
Proposition 2. Let : # K*
m . We have:
(1) If 4 | m and there is no root of unity ` # K m such that :` is
contained in a proper cyclotomic subextension of K m , then
h(:)
(2)
log 2
.
4
If 4 |% m and : is not a root of unity, then
h(:)
log 5
.
12
Proof. Assume first that 4 | m and :` is not contained in any proper
cyclotomic subextension of K m for any root of unity ` # K m . Then the
proof of the inequality h(:)(log 2)4 can be obtained as the proof of the
inequality h(:)log( p2)(2p) in Proposition 1, simply replacing the relation p | (# p &_# p ) by the stronger relation 4 | (# 2 &_# 2 ).
266
AMOROSO AND DVORNICICH
Assume now that 4 |% m and # is not a root of unity. Then the argument
of the first part of the proof of Proposition 1 gives (replacing the relation
p | (# p &_# p ) by 4 | (# 4 &_# 2 )):
2 log 2&
h(:)
2
log + |: 4 &_: 2 | v
D v|
,
6
where D=,(m). Similarly, considering : 8 &_: 4 one obtains
3 log 2&
h(:)
2
log + |: 8 &_: 4 | v
D v|
.
12
If |:| {1, the result of Schinzel [Sc 1973] quoted in the introduction gives
the better lower bound
1
- 5+1
h(:) log
.
2
2
Therefore we can assume |:| =1 (and hence |:| v =1 for all v | ). Putting
_: 2 =: 4e it and |1&e it | v = |1&e itv |, we get
h(:)
1
2
max
: (4 log 2&2 log + - 2&2 cos t v ),
12
D v|
{
2
: (3 log 2&log + - 4&4 cos 2 t v ) .
D v|
=
Let x v =cos t v ; to conclude the proof we quote the following lemma:
Lemma 4.
functions:
Let x 1 , ..., x k # (&1, 1) and consider the following real
f (x)=4 log 2&2 log + - 2&2x,
g(x)=3 log 2&log + - 4&4x 2.
Then
1
max
k
{
k
k
=
: f (x j ), : g(x j ) log 5.
j=1
j=1
LOWER BOUND FOR THE HEIGHT IN ABELIAN EXTENSIONS
Proof.
267
Let
-3
,
2
{}
=
-3
1
I = x &
<x ,
{ } 2 2=
1
I = x <x<1 .
{ }2 =
I 1 = x &1<x&
2
3
We have
f (x)2 log 2,
if x # I 1 ;
if x # I 3 ;
{ f (x)=4 log 2,
g(x)=3 log 2,
{g(x)3 log 2&log - 3,
if x # I 1 ;
if x # I 3 ;
Moreover, f and g are convex functions in I 2 . Hence,
k2
1
k1
k3
1
: f (x j ) } 2 log 2+ } f
: x j + } 4 log 2;
k j
k
k
k2 x # I
k
\
1
k
k
1
: g(x ) } 3 log 2+ } g
\k
k
k
k
1
2
j
j
+
k
: x + } (3 log 2&log - 3),
+ k
2
3
j
2 x #I
j
2
j
where k l denotes the number of j for which x j # I l . Let
F(x 0 ; y 1 , y 2 , y 3 )= y 1 } 2 log 2+ y 2 } f (x 0 )+ y 3 } 4 log 2,
G(x 0 ; y 1 , y 2 , y 3 )= y 1 } 3 log 2+ y 2 } g(x 0 )+ y 3 } (3 log 2&log - 3).
We infer that
1
max
k
{
k
k
: f (x j ), : g(x j )
j=1
min
x 0 # I2
y1 +y2 +y3 =1
j=1
=
max[F(x 0 ; y 1 , y 2 , y 3 ), G(x 0 ; y 1 , y 2 , y 3 )]=log 5. K
We now combine Propositions 1 and 2. We observe that, for primes
p13, we have
log( p2) log 5 log(112) log(52) log(72) log 2
<
<
<
<
<
.
p+1
12
12
6
8
4
268
AMOROSO AND DVORNICICH
Therefore, we define
c(m)=
log(72)
,
8
log(52)
,
6
log(112)
,
12
log 5
,
12
Theorem 1.
(1)
if 7 |3 m;
if 7 | m and
if 35 | m and
5 |3 m;
11 |3 m;
if 385 | m.
Let : # K*m . Then
If : is not a root of unity,
h(:)c(m);
(2) If 4 | m and there is no root of unity ` # K m such that :` is contained
in a proper cyclotomic subextension of K m , we have the stronger lower bound
h(:)
log 2
.
4
In particular we find, via KroneckerWeber's theorem, the result announced
in the introduction.
3. COROLLARIES
Now we state some corollaries of our main theorem. We start by giving
a lower bound for the norm of an algebraic integer:
Corollary 1. Let #{0 be an integer lying in an abelian extension L
of Q. Then, if ## is not a root of unity,
log |N LQ #| log 5
.
[L : Q]
12
Moreover the constant (log 5)12 can be replaced by c(m) if L=K m and by
(log 2)4 if L=K m and if there is no root of unity ` # K m such that :` is
contained in a proper cyclotomic subextension of K m .
LOWER BOUND FOR THE HEIGHT IN ABELIAN EXTENSIONS
Proof.
269
Let :=##. By the product formula,
[L & : Q & ]
[L & : Q & ]
log + |:| & & :
log |#| &
[L
:
Q]
[L : Q]
& |3 & |3 h(:)= :
[L & : Q & ]
log |N LQ #|
log |#| & =
.
[L : Q]
& | [L : Q]
=:
Now, apply Theorem 1. K
Let ; be a non-reciprocal algebraic number (i.e. such that ; &1 is not
a conjugate of ;) of degree D. Smyth [Sm 1971], by using tools from
complex analysis, proved that the height of ; is log % 0 Dr0.2811D,
where % 0 is the smallest Pisot's number, i.e. the real root of the polynomial
x 3 &x&1. This result is optimal since h(% 0 )=log % 0 3. Corollary 1 allows
us to reobtain, although in a slightly weaker form, Smyth's result:
Theorem (Smyth). Let ; a non-reciprocal algebraic number of degree D.
Then we have
c
h(;)
D
with c=log(72)8r0.1565.
Proof. We assume that ; is an algebraic integer (since otherwise h(;)
(log 2)D). Let f (x) # Z[x] be the minimal polynomial of ; and let p{7 be
a prime number 2D+3. Let also #= f (` p ). We start by proving that ##
is not a root of unity. Assume the contrary. Then there exists an integer j
such that f (` p )=\` pj f(` &1
p ); hence
j
`D
p f (` p )=\` p f *(` p ),
where f *(x)=x Df (x &1 ) is the reciprocal polynomial of f. Without loss
of generality, we can assume &1 j p&2. By assumption on ;, the
polynomial
g(x)=x max[D& j, 0]f (x)\x max[ j&D, 0]f *(x)
is not zero; on the other hand g(` p )=0 and g has degree bounded by
D+ |D& j | max[2D+1, p&2]< p&1,
a contradiction. The hypothesis of Corollary 1 is therefore satisfied and we
find:
p
log |N K
log(72)
Q f (` p )|
.
p&1
8
270
AMOROSO AND DVORNICICH
On the other hand, denoting by ; 1 , ..., ; D the conjugates of ;,
}
Q( ;)
p
|N K
Q f (` p )| = N Q
D
; p &1
;)
p
D
|N Q(
(
;
&1)|
2
`
max[ | ; j | p, 1].
Q
;&1
j=1
}
Hence
p
f (` p )| D(log 2+ ph(;)).
log |N K
Q
Taking into account the previous lower bound for this norm we find
1 log(72) D log 2
&
.
8
p
\ p+
Dh(;) 1&
Now, we let p Ä +. K
We remark that in the preceding proof we have used only the first bound
given in Proposition 1, namely the lower bound h(:)log(72)8 for a
non-root of unity : # K *
p with p{7.
Next we state a result about the class group of abelian extensions of Q.
Corollary 2. Let L be an abelian extension of Q of degree D and let
p be a rational prime ( possibly ramified ); let also denote by f its inertial
degree. Assume that p splits completely in a quadratic imaginary extension K
of Q contained in L and denote by $ the order of a prime over p in the class
group of L. Then
f $ log p log 5
.
D
12
Moreover the constant (log 5)12 can be replaced by c(m) if L=K m and by
(log 2)4 if L=K m , 4 | m and f =1.
Proof. Let ^ be a prime of L over p and let $ be its order in the class
group of L. Hence ^ $ =(#) for some integer # # L. Let L + the maximal real
subfield of L; since KL + =L, we have:
^{^ .
Therefore, denoting by & the finite place of L corresponding to ^,
h(## )=
f $ log p
[L & : Q & ]
log |#| & =
>0.
D
D
Now, applying Theorem 1, we find f $ log pD(log 5)12 and f $ log pD
c(m) if L=K m .
LOWER BOUND FOR THE HEIGHT IN ABELIAN EXTENSIONS
271
Assume now f =1 and L=K m for some m divisible by 4. Let ` # K m be
a root of unity and let _ # Gal(LK) be such that _(`## )=`##. Then we
have the equality of ideals (_#)(# )=(#)(_# ); since (#){(# ), it follows from
unique factorization that (_#)=(#) and hence _ must be the identity. So
:=`## is not contained in any proper subextension of K m and the second
part of Theorem 1 applies. K
Remarks. (1) The condition about the splitting of p is necessary: the
prime p=157 has inertial degree 1 in the maximal real subfield K +
79 and
log 157((79&1)2)r0.1296<(log 5)12, but (at least assuming GRH)
K+
79 has class number one (see [Wa], Tables, p. 352).
(2) In the last part of Corollary 2 we cannot avoid the hypothesis
f =1. Consider for instance the cyclotomic field L=K 84 . Then p=7 splits
as the product of 2 principal primes with ramification index 6 and inertial
degree f =2. Moreover 7 splits completely in Q(- &3), but we have
2 log 7,(84)<(log 2)4.
Corollary 2 is almost optimal. Consider the cyclotomic field K 21 of class
number one; the prime p=7 splits as the product of 2 primes with
ramification index 6 and inertial degree 1. We have log 7,(21)=(log 7)12
r0.1621 and c(21)=log(52)6r0.1527. This produces the best example
we know of an algebraic number of ``small'' height lying in an abelian
extension: :=##, where (#) is a prime of K 21 over p=7, has height
h(:)=
log 7
.
12
The complete list of cyclotomic fields having class number one is wellknown (see [MM 1976]):
K m has class number one if and only if m is one of the following twentynine numbers: 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27,
28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84.
As an application of Corollary 2, we can show in a totally elementary
way that, for any ``reasonably'' small m (say m10 4 ) which does not
belong to the previous list, the field K m has class number >1. Let p(m, 1)
be the smallest prime satisfying p#1 (mod m). A simple computation
(done by a small personal computer) shows that if m10 4 does not belong
to the previous list of twenty-nine numbers and if m{23, 31, 39, 56, 72,
then
log p(m, 1)
<
,(m)
{
log 2
,
4
c(m),
if 4 | m;
otherwise,
272
AMOROSO AND DVORNICICH
and therefore K m has class number >1. The case m=23 can be treated
directly; for m=31 we remark that the prime 2 is unramified in K 31 and
splits as a product of six primes having inertial degree 5. Since
5 log 2 log 2 log(72)
=
<
,
30
6
8
we find that also K 31 has class number >1. The cases m=39, m=56 and
m=72 can be settled in a similar way, by considering the splitting of
p=13, p=2 and p=3, respectively.
To prove the ``only if '' part of the quoted result on the class number
of cyclotomic fields, we need a non-trivial upper bound for p(m, 1), say
log p(m, 1)<0.13 } ,(m) for m>m 0 , for a ``reasonable'' and explicit
absolute constant m 0 . For large m, a much stronger inequality is known:
a celebrate (and deep) result of Linnik asserts that p(m, 1)<m L where L
is an effective constant.
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